The air pump MAY be operating normally or it may be worn out or on it's way out (seizing or something) or possibly there is a problem with the air pump plumbing (rubber hose came off, or a problem with the divert valve). I'm not there to hear it, so I can't tell you if it sounds normal. It's not usually the kind of noise that causes embarassment though, so probably there is a problem.
It will operate for about the first 2 minutes after a cold start. A few minutes after that it may kick in for about 2 or 3 seconds as the compuer completes a test of the systems.
Look for a rubber hose over by the air pump that's not connected to anything. The air pump again is above the alternator, has a metal shield behind it, and you'll see the hose coming off of it, about the size of a heater hose. As a test you could locate the connector for the air pump clutch, over near the coolant tank (pressure tank) and disconnect it to see if the noise is gone on cold starts, but I would discourage leaving it disconnected as a permanant fix.

Why the new cylinder head?
The valve stems are part of the valves, so unless you also installed new valves (?) you didn't replace the valve stems. What is the history of the head you installed? Are you sure the valve stem SEALS were replaced? Possibly someone rebuilt the head and misinstalled the seals, there are seperate intake and exhaust seals, and they could have been misinstalled on one or all of the cylinders. Also if it is a used head that was simply installed, it could still have the old seals, and/or worn valve guides.
I wouldn't really think of the plug wires as a cause for this problem, no. Possibly wrong plugs, what did you install for plugs and what gap?

Gilly